Question 18
A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown the n figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take $π = \frac{22}{7})$

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Solution

$P e r i m e t e r o f s e m i - c i r c l e = π r P_{1} = π (0.5) = \frac{π}{2} c m P_{2} = π (1) = π c m P_{3} = π (1.5) = \frac{3 π}{2} c m P_{1}, P_{2}, P_{3} a r e t h e l e n g t h s o f t h e s e m i - c i r c l e s . \frac{π}{2}, π, \frac{3 π}{2}, 2 π \dots \dots P_{1} = \frac{π}{2} c m P_{2} = π c m d = P_{2} - P_{1} = π - \frac{π}{2} = \frac{π}{2} F i r s t t e r m = P_{1} = a = \frac{π}{2} c m S_{n} = \frac{n}{2} [2 a + (n - 1) d] T h e r e f o r e, S u m o f t h e l e n g t h o f 13 c o n s e c u t i v e c i r c l e s, S_{13} = \frac{13}{2} [2 (\frac{π}{2}) + (13 - 1) \frac{π}{2}] = \frac{13}{2} [π + 6 π] = \frac{13}{2} (7 π)$
= $\frac{13}{2} \times 7 \times \frac{22}{7}$
= 143 cm

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A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A of radii 0.5, 1.0 cm, 1.5 cm, 2.0 cm, ……… as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles?